Editing Law of excluded middle (section)

== Criticisms ==
The [[Catuṣkoṭi]] (tetralemma) is an ancient alternative to the law of excluded middle, which examines all four possible assignments of truth values to a proposition and its negation. It has been important in [[Indian logic]] and [[Buddhist logic]] as well as the ancient Greek philosophical school known as [[Pyrrhonism]].

Many modern logic systems replace the law of excluded middle with the concept of [[negation as failure]]. Instead of a proposition's being either true or false, a proposition is either true or not able to be proved true.<ref>{{cite book | last = Clark | first = Keith | title = Logic and Data Bases | publisher = [[Springer-Verlag]] | date = 1978 | pages = 293–322 (Negation as a failure) | url = http://www.doc.ic.ac.uk/~klc/NegAsFailure.pdf | doi = 10.1007/978-1-4684-3384-5_11}}</ref> These two dichotomies only differ in logical systems that are not [[Completeness (logic)|complete]]. The principle of negation as failure is used as a foundation for [[autoepistemic logic]], and is widely used in [[logic programming]]. In these systems, the programmer is free to assert the law of excluded middle as a true fact, but it is not built-in ''a priori'' into these systems.

Mathematicians such as [[Luitzen Egbertus Jan Brouwer|L.&nbsp;E.&nbsp;J. Brouwer]] and [[Arend Heyting]] have also contested the usefulness of the law of excluded middle in the context of modern mathematics.<ref>
{{cite book| url = https://books.google.com/books?id=uUC30fqhdlAC&pg=PA138| title = "Proof and Knowledge in Mathematics" by Michael Detlefsen| isbn = 9780415068055| last1 = Detlefsen| first1 = Michael| date = January 1992| publisher = Routledge}}</ref>

===In mathematical logic===

In modern [[mathematical logic]], the excluded middle has been argued to result in possible [[Self-refuting idea|self-contradiction]]. It is possible in logic to make well-constructed propositions that can be neither true nor false; a common example of this is the "[[Liar paradox|Liar's paradox]]",<ref>{{Cite web |last=Priest |first=Graham |date=2010-11-28 |title=Paradoxical Truth |url=https://archive.nytimes.com/opinionator.blogs.nytimes.com/2010/11/28/paradoxical-truth/ |access-date=2023-09-10 |website=Opinionator |language=en}}</ref> the statement "this statement is false", which is argued to itself be neither true nor false. [[Arthur Prior]] has argued that [[Liar paradox|The Paradox]] is not an example of a statement that cannot be true or false. The law of excluded middle still holds here as the negation of this statement "This statement is not false", can be assigned true. In [[set theory]], such a self-referential paradox can be constructed by examining the set "the set of all sets that do not contain themselves". This set is unambiguously defined, but leads to a [[Russell's paradox]]:<ref>Kevin C. Klement, {{cite IEP |url-id=par-russ |title=Russell's Paradox }}</ref><ref>{{cite journal |first=Graham |last=Priest |title=The Logical Paradoxes and the Law of Excluded Middle |journal=The Philosophical Quarterly |volume=33 |issue=131 |year=1983 |pages=160–165 |doi=10.2307/2218742 |jstor=2218742 }}</ref> does the set contain, as one of its elements, itself? However, in the modern [[Zermelo–Fraenkel set theory]], this type of contradiction is no longer admitted. Furthermore, paradoxes of self reference can be constructed without even invoking negation at all, as in [[Curry's paradox]].{{cn|date=February 2022}}